• D. M. Gitman
  • I. V. Tyutin
  • B. L. Voronov
Part of the Progress in Mathematical Physics book series (PMP, volume 62)


We call attention to the fact that quantization includes the problem of a correct definition of quantum-mechanical observables like Hamiltonian, momentum, etc. as self-adjoint operators in an appropriate Hilbert space. This problem is nontrivial for systems on nontrivial manifolds or/and with singular interactions. A naïve treatment based on the experience in finite-dimensional algebra or even infinite-dimensional algebra with bounded operators can result in paradoxes and incorrect results. A set of such paradoxes is considered.


Hilbert Space Quantum Mechanic Momentum Operator Symmetric Operator Unbounded Operator 
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  1. 23.
    le Bellac, M.: Quantum Physics. Cambridge University Press, Cambridge (2006)zbMATHGoogle Scholar
  2. 25.
    Berezin, F.A.: The Method of Second Quantization. Academic Press, New York (1966)zbMATHGoogle Scholar
  3. 26.
    Berezin, F.A., Faddeev, L.D.: A remark on Schrödinger’s equation with a singular potential. Sov. Math. Dokl. 2, 372–375 (1961)zbMATHGoogle Scholar
  4. 27.
    Berezin, F.A., Shubin, M.A.: Schrödinger Equation. Kluwer, New York (1991)zbMATHCrossRefGoogle Scholar
  5. 31.
    Bonneau, G., Faraut, J., Valent, G.: Self-adjoint extensions of operators and the teaching of quantum mechanics. Am. J. Phys. 69, 322–331 (2001)CrossRefGoogle Scholar
  6. 32.
    Bohm, D.: Quantum Theory. Prentice-Hall, Englewood Cliffs, NJ (1951)Google Scholar
  7. 37.
    Capri, A.: Nonrelativistic Quantum Mechanics. World Scientific Publishers, Singapore (2002)Google Scholar
  8. 39.
    Cohen-Tannoudji, C., Diu, B., Laloë, F.: Quantum Mechanics. Wiley, New York (1977)Google Scholar
  9. 44.
    Davydov, A.S.: Quantum Mechanics, 2nd edn. Pergamon Press, Oxford/New York (1976)Google Scholar
  10. 48.
    Dirac, P.A.M.: The Principles of Quantum Mechanics. Clarendon Press, Oxford (1958)zbMATHGoogle Scholar
  11. 49.
    Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, Yeshiva University, New York (1964)Google Scholar
  12. 57.
    Faddeev, L.D., Yakubovsky, O.A.: Lectures on Qunatum Mechanics. Leningrad State University Press, Leningrad (1980)Google Scholar
  13. 63.
    Galindo, A., Pascual, P.: Quantum Mechanics, vols. 1 and 2. Springer (1990, 1991)Google Scholar
  14. 64.
    Gasiorowicz, S.: Quantum Physics. Wiley, New York (1974)Google Scholar
  15. 74.
    Gieres, F.: Mathematical surprises and Dirac’s formalism in quantum mechanics. Rep. Prog. Phys. 63, 1893–1931 (2000)CrossRefGoogle Scholar
  16. 75.
    Gitman, D.M., Tyutin, I.V.: Quantization of Fields with Constraints. Springer, Berlin (1990)zbMATHCrossRefGoogle Scholar
  17. 83.
    Gustafson, S.J., Sigal, I.M.: Mathematical Concepts of Quantum Mechanics. Universitext. Springer, Berlin (2003)zbMATHCrossRefGoogle Scholar
  18. 84.
    Haag, R.: Local Quantum Physics. Springer, Berlin (1996)zbMATHGoogle Scholar
  19. 91.
    Henneaux, M., Teitelboim, C.: Quantization of Gauge Systems. Princeton University Press, Princeton (1992)zbMATHGoogle Scholar
  20. 98.
    Konishi, K., Paffuti, G.: Quantum Mechanics: A New Introduction. Oxford University Press, Oxford (2009)zbMATHGoogle Scholar
  21. 104.
    Landau, L.D., Lifshitz, E.M.: Quantum Mechanics: Non-Relativistic Theory. Pergamon Press, Oxford (1977)Google Scholar
  22. 109.
    Liboff, R.L.: Introduction to Quantum Mechanics. Addison-Wesley, New York (1994)Google Scholar
  23. 112.
    Messiah, A.: Quantum Mechanics. Interscience, New York (1961)Google Scholar
  24. 128.
    Reed, M., Simon, B.: Methods of Modern Mathematical Physics, vol. I. Functional Analysis. Academic Press, New York (1980)Google Scholar
  25. 136.
    Sakurai, J.J.: Modern Quantum Mechanics. Addison-Wesley, New York (1994)Google Scholar
  26. 138.
    Schiff, L.I.: Quantum Mechanics. McGraw–Hill, New York (1955)Google Scholar
  27. 144.
    Takhtajan, L.A.: Quantum Mechanics for Mathematicians. Graduate Studies in Mathematics, 95. American Mathematical Society (2008)Google Scholar
  28. 147.
    Thirring, W.: Quantum Mathematical Physics – Atoms, Molecules and Large Systems. Springer, Berlin (2002)Google Scholar
  29. 153.
    von Neumann, J.: Mathematische Grundlagen der Quantenmechanik. Springer, Berlin (1932)zbMATHGoogle Scholar
  30. 156.
    Voronov, B.L., Gitman, D.M., Tyutin, I.V.: Constructing quantum observables and self-adjoint extensions of symmetric operators.I. Russ. Phys. J. 50(1) 1–31 (2007)Google Scholar
  31. 157.
    B.L. Voronov, D.M. Gitman, I.V.Tyutin, Constructing quantum observables and self-adjoint extensions of symmetric operators. II. Differential operators, Russ. Phys. J. 50/9 853–884 (2007)Google Scholar
  32. 158.
    Voronov, B.L., Gitman, D.M., Tyutin, I.V.: Constructing quantum observables and self-adjoint extensions of symmetric operators. III. Self-adjoint boundary conditions. Russ. Phys. J. 51(2) 115–157 (2008)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  • D. M. Gitman
    • 1
  • I. V. Tyutin
    • 2
  • B. L. Voronov
    • 2
  1. 1.Instituto de FísicaUniversidade de São PauloSão PauloBrasil
  2. 2.Department of Theoretical PhysicsP.N. Lebedev Physical InstituteMoscowRussia

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